Selberg orthogonality for half-integral weight modular forms
Shenghao Hua
TL;DR
This work extends Selberg orthogonality to half-integral weight modular forms by exploiting the Shimura lift and Waldspurger-type relations to connect Fourier coefficients with central $L$-values. Under GRH (and related conjectures) it develops a decorrelation framework for three arithmetic contexts: (i) products of Fourier coefficients of Kohnen-space forms across primes, (ii) products of central values in prime quadratic twists of GL$_2$ forms via $L$-values and automorphic periods, and (iii) analytic orders of isotropy subgroups in Tate–Shafarevich groups of elliptic curves under prime quadratic twists. The main results give upper bounds of the form $\ll (\log X)^{-m/8}$ for mixed moments, yielding $o(X/\log X)$-type Selberg orthogonality in the half-integral weight setting and nontrivial analytic savings in several arithmetic problems. This approach blends Keating–Snaith-type predictions for orthogonal families with Soundararajan’s GRH-based method and leverages equidistribution results (Duke) and BSD/GRC assumptions to control central $L$-values, toric periods, and Shafarevich–Tate invariants, highlighting the broad impact of decorrelation on QE, variance, and BSD-related phenomena.
Abstract
The Keating--Snaith conjecture for orthogonal families may be viewed as analogous to a Gaussian distribution with a negative mean, and the possibility that mixed moments resemble a composition of independent moments, these two insights were combined and applied in Lester and Radziwi{łł}'s proof of quantum unique ergodicity for half-integral weight automorphic forms, via Soundararajan's method under the Generalized Riemann Hypothesis (GRH). This observation also yields a crucial and nontrivial saving in the resolution of certain arithmetic problems. Inspired by this, we select a series of typical mixed orthogonal families of $L$-functions: $\mathrm{GL}_2$ quadratic twisted families, Gao and Zhao established a sharp upper bound by building upon Harper's method, and one can replace square-free numbers with primes in this argument. Under the assumptions of the GRH and the Generalized Ramanujan Conjecture, we present the following three arithmetic applications: i) The decorrelation of Fourier coefficients of half-integral weight modular forms, specifically, a variant of Selberg orthogonality for distinct half-integral weight modular forms. ii) The decorrelation of automorphic periods averaged over prime imaginary quadratic fields. iii) The decorrelation of the analytic orders of isotropy subgroups of Tate--Shafarevich groups of elliptic curves under prime quadratic twists.